An LED light source device comprising: an LED (12, 15, 18) including an LED element (13, 16, 19) as a light emission source; a light reflecting body configured to include a light reflecting surface (44, 45, 46, 51, 56) configured to reflect light emitted from the LED (12, 15, 18) in a predetermined direction; an LED mounting board; a circuit board; and a pair of supporting main bodies (60, 80), the LED light source device being characterized in that the light reflecting surface (44, 45, 46, 51, 56) is configured to include at least a first light reflecting surface (44, 45, 46) composed of a first hyperbolic cylindrical surface; the first hyperbolic cylindrical surface (44, 45, 46) is configured to have an inner focal line inside the light reflecting body and be obtained by moving part of a hyperbolic line (44a, 45a, 46a) in a perpendicular direction to a plane including the hyperbolic line (44a, 45a, 46a), the hyperbolic line (44a, 45a, 46a) having an outer focal point (f2, f2a, f2b) at or near a position at which the LED element (13, 16, 19) is located; the first light reflecting surface (44, 45) is configured to further include a second hyperbolic cylindrical surface; the second hyperbolic cylindrical surface is configured to have an inner focal line located at the inner focal line of the first hyperbolic cylindrical surface and be obtained by moving part of a hyperbolic line in a perpendicular direction to a plane including the hyperbolic line, the hyperbolic line having an outer focal point (f2b) at a position different from the outer focal point (f2a) of the hyperbolic line of the first hyperbolic cylindrical surface; each of the pair of supporting main bodies (60, 80) includes a heatsink portion (65, 85) and a trunk portion (70, 90), the heatsink portion (65, 85) including a groove (62, 82) configured to receive and support the LED mounting board, the trunk portion (70, 90) extending from the corresponding heatsink portion (65, 85) and holding the light reflecting body and the circuit board; the circuit board is inserted into and fit to a circuit board insertion groove provided to the light reflecting body; and the LED (12, 15) is mounted on the LED mounting board so that the irradiation direction of the LED (12, 15) is directed toward the light reflecting body. an LED (12, 15, 18) including an LED element (13, 16, 19) as a light emission source; a light reflecting body configured to include a light reflecting surface (44, 45, 46, 51, 56) configured to reflect light emitted from the LED (12, 15, 18) in a predetermined direction; an LED mounting board; a circuit board; and a pair of supporting main bodies (60, 80), the LED light source device being characterized in that the light reflecting surface (44, 45, 46, 51, 56) is configured to include at least a first light reflecting surface (44, 45, 46) composed of a first hyperbolic cylindrical surface; the first hyperbolic cylindrical surface (44, 45, 46) is configured to have an inner focal line inside the light reflecting body and be obtained by moving part of a hyperbolic line (44a, 45a, 46a) in a perpendicular direction to a plane including the hyperbolic line (44a, 45a, 46a), the hyperbolic line (44a, 45a, 46a) having an outer focal point (f2, f2a, f2b) at or near a position at which the LED element (13, 16, 19) is located; the first light reflecting surface (44, 45) is configured to further include a second hyperbolic cylindrical surface; the second hyperbolic cylindrical surface is configured to have an inner focal line located at the inner focal line of the first hyperbolic cylindrical surface and be obtained by moving part of a hyperbolic line in a perpendicular direction to a plane including the hyperbolic line, the hyperbolic line having an outer focal point (f2b) at a position different from the outer focal point (f2a) of the hyperbolic line of the first hyperbolic cylindrical surface; each of the pair of supporting main bodies (60, 80) includes a heatsink portion (65, 85) and a trunk portion (70, 90), the heatsink portion (65, 85) including a groove (62, 82) configured to receive and support the LED mounting board, the trunk portion (70, 90) extending from the corresponding heatsink portion (65, 85) and holding the light reflecting body and the circuit board; the circuit board is inserted into and fit to a circuit board insertion groove provided to the light reflecting body; and the LED (12, 15) is mounted on the LED mounting board so that the irradiation direction of the LED (12, 15) is directed toward the light reflecting body.