Answer:
Explanation:
To find the equation of the tangent to the circle 4x^2 + 4y^2 = 25 that is parallel to the line 3x + 5y + 7 = 0, we can use the following steps:
Rewrite the equation of the circle in standard form: (x - a)^2 + (y - b)^2 = r^2, where (a, b) is the center of the circle and r is the radius.
In this case, the equation of the circle is already in standard form, so we can skip this step.
Find the slope of the line 3x + 5y + 7 = 0. The slope is -3/5.
Find the slope of the tangent to the circle. The slope of the tangent will be equal to the slope of the line, which is -3/5.
Substitute the slope of the tangent and the coordinates of a point on the circle into the point-slope form of the equation of a line: y - y1 = m(x - x1), where (x1, y1) is a point on the circle and m is the slope.
In this case, we can substitute the coordinates of the center of the circle (which is (0, 0)) and the slope of the tangent (-3/5) into the point-slope form to get:
y - 0 = (-3/5)(x - 0)
Simplify to get the equation of the tangent: y = -3/5x.
Therefore, the equation of the tangent to the circle 4x^2 + 4y^2 = 25 that is parallel to the line 3x + 5y + 7 = 0 is y = -3/5x.