 Hi and welcome to the session. Let us discuss the following question. Question says, at any point x, y of a curve, the slope of the tangent is twice the slope of the line segment, joining the point of contact to the point minus 4 minus 3. Find the equation of the curve given that it passes through minus 2, 1. First of all, let us understand that slope of the tangent to a curve at point x, y is equal to dy upon dx. Also, slope of the line segment joining the points x1, y1 and x2, y2 is equal to y2 minus y1 upon x2 minus x1. This is the key idea to solve the given question. Let us now start with the solution. Now we know slope of the tangent to the curve at point x, y is dy upon dx. So we can write slope of the tangent to the curve at point x, y is equal to dy upon dx. Also, slope of the line segment joining points x, y and minus 4 minus 3 is y plus 3 upon x plus 4. Now according to the question, slope of the tangent is equal to twice the slope of the line segment joining these two points. So we can write, according to the question, dy upon dx is equal to 2 multiplied by y plus 3 upon x plus 4. Now separating the variables in this equation, we get dy upon y plus 3 is equal to 2 upon x plus 4 dx. Now integrating both the sides of this equation, we get integral of dy upon y plus 3 is equal to integral of 2 dx upon x plus 4. Now this integral is equal to log of y plus 3 and this integral is equal to 2 multiplied by log of x plus 4 plus log c, where log c represents the constant of integration. We have used this formula of integration for finding out these two integrals. Now applying this law of logarithms in this term, we get log of y plus 3 is equal to log of x plus 4 whole square plus log c. Now using this law of logarithms in right hand side of this equation, we get log of c multiplied by x plus 4 whole square is equal to log of y plus 3. Now applying this law of logarithms on both the sides of this equation, we get y plus 3 is equal to c multiplied by x plus 4 whole square. Let us name this equation as equation 1. Now equation 1 represents the equation of the curve and this curve passes through the point minus 2, 1. So we will substitute minus 2 for x and 1 for y in this equation and we get 1 plus 3 is equal to c multiplied by minus 2 plus 4 whole square. Now this further implies 4 is equal to c multiplied by square of 2. We know 4 minus 2 is equal to 2. So here we will get 2 square. Now 2 square is equal to 4. Now we get 4 is equal to 4c. Now dividing both the sides of this equation by 4, we get 1 is equal to c or we can simply write c is equal to 1. Now substituting value of c is equal to 1 in equation 1, we get y plus 3 is equal to 1 multiplied by x plus 4 whole square. Now this further implies y plus 3 is equal to square of x plus 4. So the required equation of the curve is x plus 4 whole square is equal to y plus 3. This is our required answer. This completes the session. Hope you understood the solution. Take care and keep smiling.