La circunferencia es tangente a la recta 3x - 4y = 34 en el punto (10, -1) Y también es tangente a la recta 4x + 3y = 12 en el punto (3, O). Hallar la ecuación de la circnferencia.

Answer:

La ecuación del círculo en la forma es (x - 145/12)² + (y - 109/16)² = (545/48)²

Step-by-step explanation:

La información dada son;

La primera línea tangente al círculo en el punto (10, -1) = 3·x - 4·y = 34

La segunda línea tangente al círculo en el punto (3, 0) = 4·x + 3·y = 12

Las rectas tangentes dadas en forma de pendiente e intersección, y = m·x + c son;

3·x - 4·y = 34

4·y = 34 - 3·x

y = 34/4 - 3/4·x = 8.5 - 3/4·x

Primera tangente

y = 8.5 - 3/4·x

3·y = 12 - 4·x

y = 12/3 - 4/3·x = 4 - 4/3·x

Segunda tangente

y = 4 - 4/3·x

La recta de la ecuación de los radios es perpendicular a la tangente y se encuentra con la tangente en el punto de intersección con el círculo como sigue;

La primera ecuación radial tiene pendiente, m = 1/(-3/4) = 4/3

La ecuación en forma de punto y pendiente es y - (-1) = 4/3×(x - 10)

y = 4/3×(x - 10) - 1 = 4/3·x - 40/3 - 1 = 4/3·x - 43/3

Por tanto, la ecuación del primer radio es;

y = 4/3·x - 43/3

La segunda ecuación radial tiene pendiente, m = 1/(-4/3) = 3/4

La ecuación en forma de punto y pendiente es y - 0 = 3/4×(x - 3)

y = 3/4×(x - 3) = 3/4·x - 9/4

Por tanto, la ecuación del segundo radio es;

y = 3/4·x - 9/4

Los dos radios se encuentran en el centro dado como sigue;

4/3·x - 43/3 = 3/4·x - 9/4

4/3·x - 3/4·x  = 43/3 - 9/4

7/12·x = 145/12

x = 145/12

y = 3/4·x - 9/4 = 3/4·145/12 - 9/4 = 109/16

Las coordenadas del centro = (h, k) = (145/12, 109/16)

La longitud del radio, r = √((109/16 - 0)² + (145/12 - 3)²) = 545/48

La ecuación del círculo en la forma (x - h)² + (y - k)² = r² es, por lo tanto;

(x - 145/12)² + (y - 109/16)² = (545/48)².

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